P–V–T equation of state of Na-majorite to 21GPa and 1673K

Physics of the Earth and Planetary Interiors 227 (2014) 68–75

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P–V–T equation of state of Na-majorite to 21 GPa and 1673 K

0031-9201/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.pepi.2013.11.005

⇑ Corresponding author at: V.S. Sobolev Institute of geology and mineralogy,Siberian branch of Russian Academy of Sciences, Novosibirsk 630090, Russia. Tel./fax: +7 (383) 3332792.

E-mail address: [email protected] (A.M. Dymshits).

Anna M. Dymshits a,b,⇑, Konstantin D. Litasov a,b, Anton Shatskiy a,b, Igor S. Sharygin a,b, Eiji Ohtani c,Akio Suzuki c, Nikolay P. Pokhilenko a, Kenichi Funakoshi d

a V.S. Sobolev Institute of Geology and Mineralogy, Siberian Branch of Russian Academy of Sciences, Novosibirsk 630090, Russiab Novosibirsk State University, Novosibirsk 630090, Russiac Department of Earth and Planetary Materials Science, Graduate School of Science, Tohoku University, Sendai 980-8578, Japand SPring-8, Japan Synchrotron Radiation Research Institute, Kouto, Hyogo 678-5198, Japan

a r t i c l e i n f o

Article history:Received 6 May 2013Received in revised form 6 November 2013Accepted 7 November 2013Available online 20 November 2013Edited by K. Hirose

Keywords:Na-majoriteEquation of stateX-ray diffractionHigh pressure experiment

a b s t r a c t

The P–V–T equation of state (EoS) for Na-majorite at pressures to 21 GPa and temperatures to 1673 K wasobtained from in situ X-ray diffraction experiments using a Kawai-type multi-anvil apparatus. Analyses ofthe room-temperature P–V data to a third-order Birch–Murnaghan EoS yielded ambient unit cell volume,V0 = 1476 (1) (Å3); isothermal bulk modulus, K0,300 = 181 (9) GPa; and its pressure derivative, K 00;300 ¼ 4:4(1.2). When fitting a high-temperature Birch–Murnaghan EoS using entire P–V–T data at a fixedV0 = 1475.88 Å3, K0,300 = 184 (4) GPa, K 00;300 ¼ 3:8 (6), (@K0,T/@ T)P = �0.023 (5) (GPa K�1), and a = 3.17(16) � 10�5 K�1, b = 0.16 (26) � 10�8 K�2, where a = a + bT is the volumetric thermal expansion coeffi-cient. Fitting the Mie–Grüneisen–Debye EoS with the present data to Debye temperature fixed ath0 = 890 K yielded Grüneisen parameter, c0 = 1.35 at q = 1.0 (fixed). The new data on Na-majorite werecompared with the previous data on majorite type garnets. The compression mechanism of majoritic gar-nets was described in detail. The entire dataset enabled to examine the thermoelastic properties ofimportant mantle garnets and these data will have further applications for modeling P–T conditions inthe transition zone of the Earth’s mantle using ultradeep mineral assemblages.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

Garnet is one of the most abundant mineral in the upper mantleand transition zone and can comprise up to 40 vol.% of peridotiticand up to 70 vol.% of basaltic or eclogitic lithologies (Akaogi andAkimoto, 1977; Anderson and Bass, 1984; Irifune and Ringwood,1987; Ita and Stixrude, 1992). With increasing pressure garnet be-comes progressively depleted in Al and Cr while the Si content inoctahedral site, as well as concentrations of the divalent cations(Ca, Mg, Fe) and Na, regularly increase (Fig. 1). Na admixture (upto 0.22 wt.%) was originally discovered in pyrope garnets fromeclogite xenoliths and inclusions in diamonds derived by Siberiankimberlites, as well as in ultra-high-pressure (UHP) complexesand inclusions in diamonds (Sobolev and Lavrent’ev, 1971). Theauthors suggested direct connection between the sodium concen-trations and the six-coordinated silicon that can be expressed asa function of pressure. Later, Gasparik (1989) documented manyhigh-silicon garnets, including a sample close to the end-membercomposition of Na-majorite, Na2MgSi5O12 (Na-maj). Recently,

Bobrov et al. (2008a,b) and Dymshits et al. (2013) experimentallydemonstrated that the amount of Na-maj in garnet may signifi-cantly increase with pressure. Thus, it can be used as the pressuremarker for mineral assemblages with Na-majoritic garnets at theconditions of the deep upper mantle and transition zone. More-over, due to the complex composition of natural garnets, whichcan widely vary depending on the conditions (Harte, 2010; Kiseevaet al., 2013), the study of the thermoelastic properties of end-mem-ber garnets is an important key to understand the processes occur-ring at high pressures and temperatures in the mantle. Mineralassemblages with majoritic garnet are of special importance, be-cause they can crystallize under the conditions of deep upper man-tle and transition zone (Stachel, 2001).

Methods of depth estimation with account for the compositionof majoritic garnet were developed by Stachel (2001), Collersonet al. (2010), and Simakov and Bobrov (2008). Experimentally ob-tained dependence of Na and Si contents (Na-maj component) ingarnet on pressure should be taken into account for the calculationof improved geobarometers for majorite-bearing mineral assem-blages (Harte, 2010; Kiseeva et al., 2013). Collerson et al. (2010)have derived a tools for empirical estimation of pressure in naturalgarnets based on both the coupled substitution (Na+)[1+] (Ti + [VI]-

Si)[4+] = (M)[2+] (Al + Cr)[3+], and the classic pyroxene-stoichiometrymajorite-substitution. In this equation sodium can be expressed as

Fig. 1. Na-bearing majoritic garnets compositions from inclusions in diamonds (Moore and Gurney, 1985; Davies et al., 1999; Stachel, 2001; Pokhilenko et al., 2004; Harteand Cayzer, 2007; Shatskii et al., 2010). Pressure scale based on experimental data interpreted by Stachel (2001).

A.M. Dymshits et al. / Physics of the Earth and Planetary Interiors 227 (2014) 68–75 69

Na-maj component. To improve the approach suggested by Coller-son more thermodynamic parameters of Na-maj and Na-pyroxene(Na-px) are needed. However, thermodynamic data for Na-maj aswell as its mixing properties have been poorly determined. Thecrystal structure of Na-maj has been studied at ambient conditions(Pacalo et al., 1992; Bindi et al., 2011) and at high pressures andtemperatures by atomistic modeling (Vinograd et al., 2011). Thecompressibility curve was studied at ambient temperature byHazen et al. (1994). Hazen et al. (1994) investigated compressibil-ity of synthetic Na-maj [(Na1.88Mg1.12) (Mg0.06Si1.94) Si3O12] by sin-gle-crystal X-ray diffraction using Merrill–Bassett diamond anvilcell up to 5 GPa. The obtained value of 191.5 ± 2.5 GPa for isother-mal bulk modulus differs significantly from the adiabatic bulkmodulus measured at ambient conditions using Brillouin spectros-copy for the same composition, which was 173.5 GPa (Pacalo et al.,1992). We can emphasize that crystal analyzed by Pacalo et al.(1992) have some amount of majorite that has bulk modulus closeto 163 GPa (Hunt et al., 2010). It means that the value for pure Na-maj would be higher than one obtained by Pacalo et al. (1992). Thediscrepancies between two studies for the crystals of the samecomposition were explained by Hazen et al. (1994) as the resultof the assumption for K 00;300 ¼ 4. Being greater than 5, K 00;300 reducesbulk modulus to the value below 190 GPa. However, Hazen et al.(1994) obtained only 8 experimental points at room temperaturethat is not enough for the accurate refinement of the bulk modulus.Here, we present pressure–volume–temperature relations for syn-thetic Na-maj by means of multi-anvil press experiments com-bined with X-ray diffraction at pressures to 21 GPa andtemperatures to 1673 K. A complete set of the thermoelasticparameters for P–T conditions of the transition zone are extractedusing various equations of state and discussed by comparing withthose of the previous studies.

2. Experimental procedure and sample description

The starting material was a gel of Na2MgSi5O12 prepared usingthe nitrate gelling method (Hamilton and Henderson, 1968). Thestarting material was mixed with 5 wt.% Au powder used as pres-sure marker.

In situ X-ray diffraction experiment #P187 was conducted at thePhoton Factory (Tsukuba, Japan), using a 700-tons Kawai-typemulti-anvil apparatus ‘‘MAX-III’’ installed at a bending magnetbeamline NE7A (Suzuki et al., 2011). We used 22 mm WC anvils(Tungaloy F-grade) with a truncated edge length of 3.5 mm. The

sample assembly was essentially the same with that used in(Litasov and Ohtani, 2009) but modified for the in situ study(Fig. 2). The experimental setup is explained in details in Litasovet al. (2013). It consisted of a ZrO2 pressure medium, a cylindricalLaCrO3 heater, molybdenum electrodes, and a BN sample capsule.Temperature was monitored by a W25%Re–W3%Re thermocouplewith a junction located at nearly the same position as where theX-ray path through the sample. Runs #M1127 and #S2683 wereconducted at the ‘‘SPring-8’’ synchrotron radiation facility (Hyogo,Japan), using a 1500-tons Kawai-type multi-anvil apparatus,‘‘Speed-Mk.II’’ and ‘‘SPEED-1500’’, installed at a bending magnetbeam line BL04B1 (Utsumi et al., 1998). We also used oscillationsystem installed in SPEED-Mk.II to prevent the peak disappearancefrom the diffraction (Katsura et al., 2004) because of the crystalgrowth. This system was quite successful to get ideal diffractionpattern in spite of the limitation of the oscillation angle (6�). Thedesign of cell assembly was the same with described above forRun #P187.

Experiments were performed at 3–21 GPa and 300–1673 K(Fig. 3; Table 1). The cell assembly was first compressed to desiredpress load at ambient temperature. Thereafter, we followed a com-plex P–T-path with 2 or 3 heating/decompression circles (Fig. 3)while continuously taking diffraction patterns. Typical exposuretime for collecting diffraction data were 200–400 s.

Representative X-ray diffraction patterns are shown in Fig. 4illustrating diffraction peaks of Na-maj, stishovite, and Au pressuremarker. The experimental pressures were calculated from the unitcell volume of Au using the equation of state (EoS) from(Dorogokupets and Dewaele, 2007; Sokolova et al., 2013). Typi-cally, 4–5 of the diffraction lines [(111), (200), (220), (311), and(222)] of Au were used to calculate the pressures, and about8-10 diffraction lines were used to calculate the volume ofNa-maj (Fig. 4). Refinement of peaks positions and determinationthe d-values were achieved using the XRayAnalysis software pro-vided by beam line. The refinement of weak individual and someoverlapped peaks were re-examined manually. The uncertaintiesof unit cell volume of Au, determined by least-square method, givetypically 0.1 GPa uncertainty in pressure. The unit cell volume ofNa-maj was calculated using the UnitCell software (Holland andRedfern, 1997).

The recovered samples were examined with an electron micro-probe (JEOL Superprobe JXA-8800) at Tohoku University. An accel-eration voltage of 15 kV and 10 nA specimen current was used forthe analyses. The compositions of recovered garnets were slightly

Fig. 2. Schematic illustration of the 3.5 TEL high-pressure cell used for experiments.

Fig. 3. Pressure–temperature conditions of in situ X-ray diffraction experiments.The pressures were calculated using EoS of Au (Dorogokupets and Dewaele (2007);Sokolova et al. (2013)). Dotted line covers the conditions studied by Hazen et al.(1994). Solid line illustrates Na-px–Na-maj transition (Dymshits et al., 2013).

70 A.M. Dymshits et al. / Physics of the Earth and Planetary Interiors 227 (2014) 68–75

deviated from stoichiometry to be Na1.90Mg1.00Si5.00O12 due to theloss of sodium under electron beam. The consistency of synthe-sized garnet compositions were confirmed by relevant consistencyof calculated P–V–T data in different runs.

3. Results and discussion

3.1. Pressure–volume data at room temperature

The unit cell parameters of Na-maj obtained after experiment#P187 at ambient conditions are a = 11.3855 (4) Å andV0 = 1475.88 (16) Å3. The cell volume is slightly larger than that ob-tained by single crystal observations, 1472.5 Å3 (Bindi et al., 2011).The cell parameters reported by Hazen et al. (1994) and Pacaloet al. (1992) are around 1485.5 Å3 that is about 0.7% larger thanthat obtained in the present study. The unit cell parameter andvolume of Na-maj are lower than those for any other garnetend-members (Milman et al., 2001; Chopelas, 2006). As it wasdemonstrated by Bindi et al. (2011), Na and Mg are disordered atthe X site, whereas both Y (octahedral) and Z (tetrahedral) sites

are occupied by silicon. Passing from pure pyrope to pure Na-majwe observe the substitution of Si for Al in Y site. The differencein their size is significant that may cause decrease of the Y site dis-tances relative to pyrope (1.79 Å and 1.85 Å, respectively). At thesame time, transition from an X site fully occupied with Mg to amixed (Na, Mg) population results in the slight increase of theX-O distances (from 2.28 Å to 2.31 Å). As a result, a decrease ofthe unit-cell parameters is observed from pure pyrope to Na-majorite.

The pressure–volume relations have been determined at 300 Kfrom 0 to 17 GPa by fitting the experimental data to a third-orderBirch–Murnaghan (BM) EoS:

PðV ;300Þ ¼ 1:5K0;300V0;300

VP;300

� �7=3

� V0;300

VP;300

� �5=3" #

� 1� 0:75 4� K 00;300

� � V0;300

VP;300

� �2=3

� 1

!" #ð1Þ

where V0,300, K0,300 and K 00;300 are unit-cell volume, isothermal bulkmodulus and its pressure derivative at ambient condition, respec-tively. When K 0T is fixed to 4, the fitting to BM EoS givesK0,300 = 184 (3) and V0 = 1476 (1) Å3. Adiabatic bulk modulus mea-sured at ambient conditions using Brillouin spectroscopy gives rel-atively low value, KS,0 = 173.5 GPa (Pacalo et al., 1992) that is closeto grossular garnet (Gréaux et al., 2011). It appears to connect withMg4Si4O12 admixture in that garnet (Pacalo et al., 1992; Hazen et al.,1994). The fitting of Eq. (1) to the data with all parameters variableyields V0 = 1476 (1) Å3, K0,300 = 181 (9) GPa, and K 00;300 ¼ 4:4 (1.2)and gives the values of K0,300 = 180 (5) GPa and K 00;300 ¼ 4:5 (9) forthe V0 fixed at 1475.88 Å3. Both calculations are closely agree witheach other. Since the fitted V0,300 is the same as the measured valueswithin the errors it is suitable to fix V0,300 at 1475.9 Å3 in the all fol-lowing fittings procedures.

The obtained P–V relations differ slightly from those previouslyreported for garnets referred in literature as Na-maj (Table 2). Thediscrepancy between the parameters obtained by Hazen et al.(1994) and in the present work can be explained by insufficiencyof experimental data points for accurate refinement. The presentpressure–relative volume relation of Na-maj agrees well withthose of Hazen et al. (1994) (Fig. 5). It means that the real bulkmodulus of the garnet obtained by Hazen et al. (1994) can belower.

Table 1Unit cell parameters of Na-majorite at various P–T conditions.

VAu (Å3) P (GPa) T(K) aNa-maj (Å) VNa-maj (Å3) VAu (Å3) P (GPa) T(K) aNa-maj (Å) VNa-maj (Å3)

Run P187 62.96 (2) 19.3 (1) 873 11.1011 (3) 1367.88 (12)65.66 (3) 14.0 (1) 1473 11.2416 (3) 1420.65 (13) 62.82 (2) 18.6 (1) 673 11.0940 (3) 1365.52 (12)65.68 (4) 14.0 (1) 1473 11.2414 (4) 1420.59 (12) 62.69 (3) 17.8 (1) 473 11.0900 (3) 1363.94 (12)65.80 (3) 13.6 (2) 1473 11.2481 (3) 1423.11 (13) 62.60 (3) 17.0 (1) 300 11.0920 (3) 1364.70 (11)65.20 (1) 14.1 (1) 1273 11.2271 (3) 1415.14 (12) 63.01 (3) 15.4 (1) 300 11.1165 (3) 1373.48 (12)63.87 (4) 12.1 (1) 300 11.1670 (3) 1392.47 (12) 64.26 (3) 17.2 (2) 1273 11.1640 (4) 1391.68 (13)65.35 (2) 12.2 (1) 1073 11.2460 (3) 1420.67 (12) 64.87 (2) 16.5 (1) 1473 11.2000 (4) 1405.91 (15)65.62 (1) 12.8 (1) 1273 11.2460 (3) 1422.31 (12) 64.48 (4) 16.4 (1) 1273 11.1820 (4) 1398.06 (14)64.29 (4) 11.7 (2) 473 11.1900 (4) 1401.17 (13) 64.23 (5) 15.9 (2) 1073 11.1681 (4) 1393.05 (13)65.77 (4) 11.0 (1) 1073 11.2680 (3) 1430.50 (12) 64.03 (4) 15.3 (1) 873 11.1580 (4) 1389.64 (13)65.87 (4) 12.0 (1) 1273 11.2589 (3) 1427.23 (12) 63.86 (4) 14.6 (1) 673 11.1520 (4) 1387.16 (13)63.20 (2) 14.6 (1) 300 11.1210 (3) 1375.27 (12) 63.70 (5) 13.8 (1) 473 11.1486 (4) 1385.63 (13)64.36 (4) 15.5 (2) 1073 11.1860 (4) 1399.62 (15) 63.54 (2) 13.3 (1) 300 11.1458 (4) 1384.62 (13)64.96 (4) 14.8 (1) 1273 11.2131 (3) 1409.89 (12) Run S268365.55 (5) 14.4 (2) 1473 11.2351 (3) 1418.19 (12) 66.71 (3) 11.1 (1) 1473 11.2992 (3) 1442.70 (13)66.18 (2) 13.9 (3) 1673 11.2559 (3) 1426.43 (13) 66.53 (2) 10.2 (1) 1273 11.2900 (4) 1439.25 (15)65.64 (2) 14.1 (2) 1473 11.2396 (3) 1419.88 (12) 66.19 (3) 9.8 (1) 1073 11.2820 (4) 1436.07 (16)65.19 (2) 14.1 (1) 1273 11.2252 (3) 1414.44 (12) 65.85 (3) 9.4 (1) 873 11.2617 (3) 1427.98 (12)64.34 (3) 12.9 (1) 673 11.1900 (3) 1400.98 (12) 65.56 (3) 8.9 (1) 673 11.2611 (3) 1427.48 (13)64.07 (3) 12.5 (1) 473 11.1760 (3) 1396.09 (12) 65.10 (2) 7.8 (1) 300 11.2350 (3) 1418.19 (12)66.18 (2) 8.4 (2) 873 11.2860 (4) 1437.65 (13) 64.92 (2) 8.4 (1) 300 11.2270 (3) 1414.95 (12)65.97 (4) 7.6 (1) 673 11.2730 (3) 1432.58 (12) 64.72 (2) 9.1 (1) 300 11.2092 (3) 1408.36 (12)65.65 (2) 7.2 (1) 473 11.2640 (3) 1429.10 (12) 64.38 (1) 10.2 (1) 300 11.1988 (3) 1404.22 (12)67.33 (3) 5.3 (3) 873 11.3440 (3) 1460.08 (13) 64.48 (2) 11.0 (1) 473 11.2059 (4) 1406.84 (14)66.93 (3) 5.0 (2) 673 11.3310 (3) 1454.93 (13) 64.64 (2) 11.8 (1) 673 11.2063 (4) 1407.31 (12)66.74 (4) 4.1 (2) 473 11.3200 (4) 1450.62 (14) 64.87 (3) 12.4 (1) 873 11.2120 (3) 1409.33 (12)66.67 (2) 3.1 (1) 300 11.3230 (4) 1451.65 (14) 65.09 (4) 13.1 (1) 1073 11.2154 (3) 1410.61 (13)67.93 (1) 0 300 11.3855 (4) 1475.88 (16) 65.31 (3) 13.7 (1) 1273 11.2250 (3) 1414.36 (12)

Run M1127 65.58 (4) 14.3 (1) 1473 11.2390 (3) 1419.42 (12)64.00 (3) 20.7 (1) 1673 11.1442 (3) 1384.00 (12) 65.46 (3) 14.6 (1) 1473 11.2326 (3) 1417.23 (12)63.67 (2) 20.6 (1) 1473 11.1230 (3) 1379.66 (12) 65.35 (4) 15.0 (1) 1473 11.2295 (3) 1416.02 (12)63.34 (2) 20.5 (1) 1273 11.1180 (3) 1374.24 (12) 65.51 (3) 15.8 (1) 1673 11.2298 (3) 1416.17 (12)63.12 (2) 20.0 (1) 1073 11.1090 (3) 1371.01 (12) 65.24 (4) 13.7 (1) 1673 11.2622 (3) 1428.47 (12)

Numbers in parenthesis represent the relative error calculated for a, V and P. Pressure was calculated from the Eos of Au (Dorogokupets and Dewaele, 2007; Sokolova et al.,2013).

Fig. 4. Examples of X-ray diffraction pattern collected for Na-maj in Na-maj stability field (a) and Na-px stability field (b). Px – Na-px; Gt – Na-maj; St – stishovite; Au – gold;X – unidentified picks.

A.M. Dymshits et al. / Physics of the Earth and Planetary Interiors 227 (2014) 68–75 71

Table 2The results of the fitting by the HTBM EoS for Na-majorite, compared to previous works on other majorite-type and some other garnets.

V0 (Å3) K0,300 (GPa) K 00;300 (@K0,T/@T) (GPa K�1) a0,300 (10�5 K�1) a (10�5 K�1) b (10�8 K�2)

1) Na-maj 1475.9a 182 (1) 4a �0.025 (4) 3.27 (17) 3.20 (14) 0.24 (20)1475.9a 177 (1) 5a �0.031 (3) 3.48 (21) 3.31 (14) 0.58 (24)1475.9a 184 (4) 3.8 (6) �0.023 (5) 3.23 (15) 3.18 (16) 0.18 (21)1475.9a 180a 4.3 (1) �0.026 (3) 3.37 (15) 3.27 (13) 0.32 (24)1476 (1) 186 (6) 3.6 (7) �0.023 (5) 3.22 (18) 3.17 (16) 0.16 (26)

300K BM EoS1476 (1) 184 (3) 4a

1475.9a 180 (5) 4.5 (9)1476 (1) 181 (9) 4 (1)

2) NaMaj95Maj5 173.53) NaMaj94Maj6 1485.5 (3) 192 44) Ca-maj 1547.0 (3) 1655) Maj 1520.0 165 (3) 4.2 (3)6) Maj garnet 1574.1 173 (1) 4a �0.022 (5) 2.0 (3) 1.0 (5)7) Prp50Maj50 166 (3) 4.2 (3) �0.022 (2)8) Prp100 1500.4 (2) 167 4.6 �0.021 (9) 2.58 (20) 1.02 (46)9) Alm86Prp7Spe7 1539.8 177 4a �0.032 3.1 (7)10) And 1754.1 158 (2) 4 3.16 (25)

1) Na-maj – Na2MgSi5O12; 2) Na-maj – Pacalo et al. (1992); 3) Na-maj and 4) Ca-maj – Ca0.49Mg3.51Si4O12, Hazen et al. (1994); 5) Maj – Mg4Si4O12, Stixrude and Lithgow-Bertelloni (2005); 6) majorite synthesized from natural MORB, Nishihara et al. (2005); 7) Sinogeikin and Bass (2002); 8) Prp – Mg3Al2Si3O12,Zou et al. (2012); 9) Alm86-

Prp7Spe7 – almandine-spessartine solid solution, Fan et al. (2009); 10) And –Ca3Al0.03Fe1.97 Si3O12, Pavese et al. (2001). The bold line represents the calculation with the moreaccurate parameters.

a Fixed values during the fitting.

Fig. 5. Comparison of room temperature compression (V/V0) data and compress-ibility curve for Na-maj (this study) and other garnets: pyrope (Zou et al., 2012),calcium-bearing majorite (Ca0.49Mg2.51)(MgSi)Si3O12 and Na-maj (Na1.88Mg1.12(-Mg0.06Si1.94)Si3O12 (Hazen et al., 1994), synthetic MORB majorite (Nishihara et al.,2005).

72 A.M. Dymshits et al. / Physics of the Earth and Planetary Interiors 227 (2014) 68–75

Milman et al. (2001) showed that the bulk modulus of garnet isstrongly affected by the bulk modulus of the dodecahedra, whilecompressibility of other individual polyhedra displays no correla-tion with the compressibility of the structure as a whole. If so,Na-maj would have the smallest bulk modulus of all silicate gar-nets, as a phase with a predicted dodecahedral bulk modulus ofapproximately 70 GPa (Hazen et al., 1994). In fact Na-maj has thelargest bulk modulus among the silicate garnets (Fig. 5). Thisbehavior must reflect the all-mineral framework of Na-maj withvery small cell volume and silicon in the octahedral position. Thus,we conclude that not only the dodecahedral sites, but also thebehavior of the garnet framework and relative sizes of the 8- and6-coordinated cations, control garnet compression. The octahedralsite in Na-maj is quite small (1.79 Å) and contains only silicon incomparison to the pyrope (1.85 Å) or majorite (1.88 Å). The small

and highly charged octahedra share four edges with the dodecahe-dra and thus restrict the volume of the large and low chargeddodecahedra. In spite Na-maj has a large average X-cation radius(RNa = 1.07 Å) its dodecahedral volume is relatively small(V = 21.23 and 21.26 Å3) (Bindi et al., 2011).

As it was shown by Milman et al. (2001) there are two majorcompression mechanisms for garnets. One of them is the bondcompression, another is the bond bending. The more efficient com-pression can take place in XO8 dodecahedra and YO6 octahedra andwould appear as a result of polyhedra rotation. Pacalo et al. (1992)suggested that XO8 polyhedra act as braces and controls theamount of rotation between tetrahedra and octahedra within thecorner-linked chains. In case of pyrope XO8 cite is not filled upand polyhedra within the corner-linked chains can rotate freelyto accommodate applied stress. In case of Na-maj the dodecahedralsite is filled up and rotational freedom is minimized. Such relationsbetween the XO8 and YO6 sites provide evidence for comparativelymore rigid structure. As a result, Na-maj with all octahedral sitesoccupied by silicon has the largest value of the bulk modulusamong garnets. It would be interesting to study compressibilityof Li-majorite expressed by Yang et al. (2009). That phase has smal-ler cell volume (1430 Å3) and X–O distance (2.26 Å) but the sameYO6 polyhedra fully occupied by silicon.

3.2. P–V–T data and thermoelastic parameters

Pressure–volume–temperature data were used to determinethe thermoelastic properties of Na-maj with two different ap-proaches: the high-temperature Birch–Murnaghan (HTBM) EoSand the Mie–Grüneisen–Debye (MGD) EoS.

The third-order Birch–Murnaghan EoS is given by followingexpression for P(V,T):

PðV ; TÞ ¼ 1:5K0;TV0;T

VP;T

� �7=3

� V0;T

VP;T

� �5=3" #

� 1� 0:75 4� K 00;300

� � V0;T

VP;T

� �2=3

� 1

!" #ð2Þ

In the HTBM EoS the temperature effect on K0,T can be expressed asa linear function of temperature, temperature derivative (@ K0,T/@T)P

and K0,300 as follows:

A.M. Dymshits et al. / Physics of the Earth and Planetary Interiors 227 (2014) 68–75 73

K0;T ¼ K0;300 þ@K0;T

@T

� �P

� ðT � 300Þ ð3Þ

The temperature dependence of the volume at ambient pressureV0,T can be expressed as a function of the thermal expansion atzero-pressure, a0,T = a + bT:

V0;T ¼ V0;300 expZ T

300a0;T dT

� �

¼ V0;300 exp aðT � 300Þ þ 12

bðT2 � 3002Þ� �

ð4Þ

In this approach we can obtain six parametersV0;K0;300;K

00;300; ð@K0;T=@TÞ; a and b by a least squares fit. The calcu-

lated parameters are listed in Table 2. The results of the fit of Na-maj are compared to previous studies on majorite-type garnetsfor a set of different fixed K 00;300 values (Table 2). When there is noconstraint applied on the elastic parameters, fitting of Eq. (2) givesV0 = 1475.4 (1.2) Å3, K0,300 = 186 (6) GPa,K 00;300 ¼ 3:6 ð7Þ; ð@K0;T=@TÞ ¼ �0:023 (5) GPa K�1, a = 3.17 (16)� 10�5 K�1, and b = 0.16 (26) � 10�8 K�2 that are of the same orderof those obtained with the fixed V0 = 1475.9 Å3. The bulk modulusand its pressure derivative agree well with the values fitted by roomtemperature BM EoS taking the uncertainties into account. In thiscalculation the uncertainties in the bulk modulus and its first pres-sure derivative are quite large. This indicates that our data are notsufficient to constrain all the elastic parameters at the same timeduring fitting. Thus, fitting the HTBM EoS with the V0 = 1475.9 Å3

(Fig. 6) yields more realistic parameters: K0,300 = 184 (3) GPa,K 00;300 ¼ 3:8ð6Þ; ð@K0;T=@TÞ ¼ �0:023 (5) GPa K�1, a = 3.18(16) � 10�5 K�1, and b = 0.18 (21) � 10�8 K�2. The value of thermalexpansion is close to andradite.

Studies with laser induced phonon spectroscopy (Brillouinscattering and impulsively stimulated scattering) on garnets athigh-pressures generally result in K 00;300 � 4, whereas studies withultrasonic interferometry at high-pressures corresponds to highervalues for K 00;300 � 5—6:5 (e.g. (Gwanmesia et al., 2000; Sinogeikinand Bass, 2002). If K 00;300 is fixed to 4.0 we obtain K0,300 = 182.1(9),(@K0,T/@T) = �0.025 (3) GPa K�1 and a = 3.20(14) � 10�5 K�1 andb = 0.24 (20) � 10�8 K�2. The bulk modulus measured at ambientconditions using Brillouin spectroscopy gives the value close to

Fig. 6. P–V–T relations of Na-maj obtained from the present study V0 = 1475.9 Å3;K0,300 = 184 (3) GPa, K 00;300 ¼ 3:8ð6Þ; ð@K0;T=@TÞ ¼ �0:023 (5) GPa K�1, a = 3.18(16) � 10�5 K�1, and b = 0.18 (21) � 10�8 K�2, where a = a + bT is the volumetricthermal expansion coefficient The solid lines represent isothermal compressioncurves at various temperatures calculated by using the yielded thermoelasticparameters of the present study. For comparison, open circles are plotted afterHazen et al. (1994) at 300 K.

our fitting with K 00;300 fixed at 5.0. Fig. 7 shows that the tempera-ture dependences of bulk modulus (@ K0,T/@T)P for Na-maj at anyK 00;300 is much higher than for majorite or pyrope and similar toalmandine type garnet at K 00;300 ¼ 5:0. Moreover the present syn-thetic Na-maj softens faster against temperature compared toother garnets. The value of (@K0,T/@T) is moderately affected bythe variations of K 00;300 and varies from �0.023 to �0.031 GPa K�1.However, the K0,300 values changes significantly because of thestrong correlation between the bulk modulus and K 00;300 (Table 2).Thus, it is reasonable to suggest that a consistent set of parameterscan be obtained by fixing all values from the room temperature BMEoS, which gives a0,T = 3.31 (20) � 10�5 K�1 and (@K0,T/@T) = �0.029(3) GPa K�1. In reality the fitting with the only V0 fixed and other 5parameters variables provided the minimum root mean square(RMS) misfit for pressure values (Pobs � Pcal). The thermal expan-sion a0,T derived from that fitting are larger than that for other maj-orite type garnets.

In the MGD EoS (Jackson and Rigden, 1996) the pressure is de-scribed by the sum of the static pressure at room temperature P(V,300) and the thermal pressure DPth(V,T).

PðV ; TÞ ¼ PðV ;300Þ þ DPthðV ; TÞ ð5Þ

The third order Birch–Murnaghan equation (Eq. (1)) with V0 fixedduring the calculations and Mie–Grüneisen relations are used toexpress the static pressure P(V,300) and the thermal pressureDPth(V,T), respectively.

The thermal pressure is a function of the Grüneisen parameter cand the thermal energy Eth(V,T), that can be estimated using aDebye model:

DPThðV ; TÞ ¼cðV ; TÞ

V½EthðV ; TÞ � EthðV ; T0Þ� ð6Þ

EthðV ; TÞ ¼9nRT

ðh=TÞ3Z h=T

0

x3

ex � 1dx ð7Þ

where h is the Debye temperature, n = 20 is the number of atoms inthe formula unit, R is the gas constant. The volume dependence ofthe Debye temperature and Grüneisen parameter are described byfollowing equations, where q is the dimensionless power modeparameter:

Fig. 7. Isothermal bulk modulus K0,T against temperature. Solid lines represent Na-maj data for different fixed values of K 00;300 (4.0, 4.5 and 5.0) as well as red dashedline for no constraint on the elastic parameters (K0,T = 184 GPa and K0,300 = 3.8).Circles and squares symbolize the previous studies by (Hunt et al., 2010). (Forinterpretation of the references to colour in this figure legend, the reader is referredto the web version of this article.)

Fig. 8. Thermal pressure (DPth) vs. cell volume for Na-majorite. Solid lines arethermal pressures calculated using MGD EoS fit to the present data. Symbols aresame as in Fig. 6.

74 A.M. Dymshits et al. / Physics of the Earth and Planetary Interiors 227 (2014) 68–75

h ¼ h0 expc0 � c

q

� �ð8Þ

c ¼ c0vv0

� �q

ð9Þ

In this approach, six parameters V0;KT0;K0T0; c0; h0 and q can be

determined by the fitting of the P–V–T data. We recognized thatsome scattering in the present P–V–T data and limited coverage oflow-P high-T region of P–T diagram (Fig. 3) related to phase transi-tion from Na-maj to Na-px make it difficult to constrain all sixparameters by the simultaneous fitting. Therefore, fitting have beencarried out with h0 fixed at 890 K. This value was calculated fromsound velocities using the following equations based on Debye’s lat-tice vibrational model (Poirier, 2000):

h ¼�hkB

6p2nZV

� �1=3

ð10Þ

where ⁄ is the Planck’s constant (⁄ = h/2p), kB is the Boltzmann’sconstant, Z = 8 is the number of chemical formula in unit cell, V isthe unit cell volume and V is the Debye average velocity, reportedby Pacalo et al. (1992). Probably for orthosilicates measured calori-metric Debye temperature values are near 950 K and calculatedelastic h0 would be lower (Kieffer, 1979). For example, the elastich0 for grossular and pyrope are 821 and 794 K, respectively, whilecalorimetric h0 calculated from the heat capacity would be closeto 1000 K (Kieffer, 1980). The h0 value may affect the fitted otherparameters in MGD EoS fittings. To check the possible dependencewe decided to show fitted parameters for h0 = 1000 K. However,when h0 is varied by ±100 K, the results still agree within the errorbars (see below). Parameter q was also fixed at 1, because for a widerange of materials, the volume dependence of Grüneisen parameteris consistent with q equal to 1 (Stixrude and Bukowinski, 1990) andthe unit-cell volume and bulk modulus of garnets (i.e. grossular) arealmost unaffected when q values vary from 0 to 1.4 (Gréaux et al.,2011).

The results of the MGD EoS fit are summarized in Table 3 andFig. 8. From the simple thermodynamic identity (@KT/@T)V = (@KT/@T)P + aKT(@KT/@T)T and using the data at ambient conditions(KT = 184 GPa, K 00;300 ¼ ð@KT=@TÞT ¼ 3:8; ð@KT=@TÞP ¼ �0:023 GPa K�1

and a = 3.23 � 10�5 K�1) we obtain (@KT/@T)V = 0. As it was shownby Wang et al. (1998) the fact that (@KT/@T)V = 0 indicates thatthermal pressure is independent on volume and thereforeDPth(V,T) = DPth(V0,T). This is better illustrated in Fig. 8, where ther-mal pressures remain essentially constant over a wide range of vol-ume and temperature. As it can be seen from the Fig. 8 this fittingleads to almost flat trends so that DPth remains independent of cellvolume. The solid lines in Fig. 8 represents theoretical thermalpressure obtained by fitting the Eqs. (6) and (7) with experimentaldata while the points demonstrate observed values of thermalpressure. The observed values of DPth were obtained by subtractingP(V,300) calculated using the fitted BM EoS at 300 K from observedP(V,T) in the experiment. The close agreement between the calcu-lated and the observed values of the thermal pressure indicatesthat the parameters of the equation of state are accurate.

Table 3Thermoelastic parameters of Na-majorite using Mie–Grüneisen–Debye equation ofstate

V0 (Å3) 1475.9b 1475.9b

K0,300 (GPa) 184b 184b

K 00;300 3.8b 3.8b

c0 1.35 (1) 1.37 (1)q 1b 1b

h (K) 890a 1000b

a Calculated from elastic model (Pacalo et al., 1992).b Fixed.

The V0 = 1475.9 Å3, KT = 184 GPa and K 0T ¼ 3:8 were fixed duringthe fitting MGD EoS to present data as more accurate parametersobtained in the HTBM EoS. Similarly to the HTBM fitting this ap-proach leads to the minimum RMS misfit of delta pressure (Pobs–Pcal) and corresponds to more accurate calculation. We performedtwo independent series of calculations. When MGD EoS is fitted tothe present P–V–T data at fixed q = 1 and h0 = 890 K, we obtainedc0 = 1.35 (1). Fitting the present data at h0 = 1000 K yieldsc0 = 1.37 (1). Thus, the c0 is almost unaffected by changes in h0 val-ues up to 200 K. The obtained values of c0 = 1.35 � 1.37 (q = 1) arein good agreement with the value of c0 = 1.41 reported by ab initiosimulation for Mg-majorite (Stixrude and Lithgow-Bertelloni,2005). They are also similar to c0 for other garnet end-members,e.g., pyrope, (Zou et al. (2012)) and grossular, (Gréaux et al.(2011)). The value of the Grüneisen parameter is also typical formantle phases, c0 = 1.0-1.6 (e.g. Poirier, 2000) so the parametersobtained by fitting of MGD EoS to the present data seem to bereliable.

4. Summary and conclusions

Using synchrotron X-ray diffraction and Kawai-type multi-anvilapparatus, P–V–T measurements on Na-maj have been carried outat pressures between 3 and 21 GPa and temperatures up to 1673 K.Previous data on Na-maj were limited by studies at ambientconditions.

The fit of the present P–V–T data to the HTBM EoSyielded V0 = 1475.9 Å3, K0,300 = 184 (4) GPa, K 00;300 ¼ 3:8 ð6Þ;ð@K0;T=@TÞp ¼ �0:023 (5) GPa K�1, and parameters for thermalexpansion coefficient (a = a + bT): a = 3.18 (16) � 10�5 K�1 andb = 0.18 (21) � 10�8 K�2. Fitting of the present data to the MGDEoS at h0 = 890 K and q = 1 yields c0 = 1.35 (1). On the basis of thosestudies we adopt K 00;300 close to 3.8–4.4 and propose the bulkmodulus of K0,300 = 180–184 GPa and Grüneisen parameter ofc0 = 1.35–1.37.

Finally, the obtained values of the thermoelastic parameters forNa-maj are higher than for any other garnets, while the cell volumeshows the lowest value. Along with previous measurements on thehigh pressure minerals, obtained thermoelastic parameters of Na-maj (and further for other garnet end-members, e.g., knorringite)will be used for estimating pressures in natural majoritic garnets.

A.M. Dymshits et al. / Physics of the Earth and Planetary Interiors 227 (2014) 68–75 75

Acknowledgments

We thank the reviewers for critical comments and suggestions.Andrey Gorbachev is thanked for drafting (Fig. 2). Luca Bindi isthanked for crystallochemical data of Na-maj. This work was sup-ported by the Ministry of Education and Science of Russian Feder-ation (project No 14.B25.31.0032), integration project of SiberianBranch of Russian Academy of Science No 97 for 2012-2014, Rus-sian Foundation for Basic Research (Grants Nos 12-05-33008-aand 12-05-31351-mol-a), grant-in-Aid for Young Scientists No21684032 and grant-in-Aid for Scientific Research on InnovativeAreas No 20103003 from Japan Society for Promotion of Science.The project was conducted as a part of the Global Center-of-Excel-lence Program ‘Global Education and research Center for Earth andPlanetary dynamics’ at Tohoku University. Experiments were con-ducted under ‘SPring-8’ general research proposals Nos2011B1091, 2012A1416, 2012B1289 and Photon Factory No2012G031.

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